Pepper ambiguity - Revision history

BudjarnLambeth: todo improve synopsis

2025-04-13T10:55:40Z

todo improve synopsis

← Older revision		Revision as of 10:55, 13 April 2025
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	[[Category:EDO theory pages]]		[[Category:EDO theory pages]]
	[[Category:Terms]]		[[Category:Terms]]

			{{todo\|improve synopsis\|text=add a non-mathy paragraph at the start}}

Contribution at 22:18, 7 September 2024

2024-09-07T22:18:42Z

← Older revision		Revision as of 22:18, 7 September 2024
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Contribution at 15:21, 13 March 2023

2023-03-13T15:21:07Z

← Older revision		Revision as of 15:21, 13 March 2023
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	We may also define the mean ambiguity for ''N'' with respect to ''s'' by taking the mean of ambig(''N'', ''q'') for all members ''q'' of ''s''.		We may also define the mean ambiguity for ''N'' with respect to ''s'' by taking the mean of ambig(''N'', ''q'') for all members ''q'' of ''s''.
			{\| class="wikitable"
			\|+
			!odd-limit
			!list of EDOs with decreasing relative error
			\|-
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			\|1, 2, 3, 9, 16, 18, 31, 55, 129, 147, 183, 279, 436, 3513, 4349, 6850, 9934, 15112, 16808
			\|}

	== See also ==		== See also ==

Fredg999 category edits: Removing from Category:Measure using Cat-a-lot

2023-02-26T06:34:33Z

Removing from Category:Measure using Cat-a-lot

← Older revision		Revision as of 06:34, 26 February 2023
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	[[Category:EDO theory pages]]		[[Category:EDO theory pages]]
	[[Category:Terms]]		[[Category:Terms]]
	~~[[Category:Measure]]~~

Xenwolf: add odd limit after OEIS links

2020-12-11T15:26:18Z

add odd limit after OEIS links

← Older revision		Revision as of 15:26, 11 December 2020
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	Given a finite set s of positive rational numbers, the maximum value of ambig(''N'', ''q'') for all ''q'' ∈ ''s'' is the '''Pepper ambiguity''' of ''N'' with respect to ''s''. If the set ''s'' is the ''L'' odd limit [[tonality diamond]], this is the ''L''-limit Pepper ambiguity of ''N''. Lists of ''N'' of decreasing Pepper ambiguity can be found on the [[On-Line Encyclopedia of Integer Sequences]]:		Given a finite set s of positive rational numbers, the maximum value of ambig(''N'', ''q'') for all ''q'' ∈ ''s'' is the '''Pepper ambiguity''' of ''N'' with respect to ''s''. If the set ''s'' is the ''L'' odd limit [[tonality diamond]], this is the ''L''-limit Pepper ambiguity of ''N''. Lists of ''N'' of decreasing Pepper ambiguity can be found on the [[On-Line Encyclopedia of Integer Sequences]]:

	* [[OEIS: A117554]]		* [[OEIS: A117554]] — 5-odd-limit
	* [[OEIS: A117555]]		* [[OEIS: A117555]] — 7-odd-limit
	* [[OEIS: A117556]]		* [[OEIS: A117556]] — 9-odd-limit
	* [[OEIS: A117557]]		* [[OEIS: A117557]] — 11-odd-limit
	* [[OEIS: A117558]]		* [[OEIS: A117558]] — 13-odd-limit
	* [[OEIS: A117559]]		* [[OEIS: A117559]] — 15-odd-limit

	We may also define the mean ambiguity for ''N'' with respect to ''s'' by taking the mean of ambig(''N'', ''q'') for all members ''q'' of ''s''.		We may also define the mean ambiguity for ''N'' with respect to ''s'' by taking the mean of ambig(''N'', ''q'') for all members ''q'' of ''s''.

Xenwolf: added relative error as see also; within the text, the reference wasn't really precise (v is a measure - or equal to - abs(relative error))

2020-12-11T15:19:46Z

added relative error as see also; within the text, the reference wasn't really precise (v is a measure - or equal to - abs(relative error))

← Older revision		Revision as of 15:19, 11 December 2020
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	We may also define the mean ambiguity for ''N'' with respect to ''s'' by taking the mean of ambig(''N'', ''q'') for all members ''q'' of ''s''.		We may also define the mean ambiguity for ''N'' with respect to ''s'' by taking the mean of ambig(''N'', ''q'') for all members ''q'' of ''s''.

			== See also ==
			* [[Relative error]]

	[[Category:EDO theory pages]]		[[Category:EDO theory pages]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Measure]]		[[Category:Measure]]

FloraC: Categories

2020-12-11T09:48:31Z

FloraC: Cleanup and improve readability

2020-12-11T09:47:03Z

Cleanup and improve readability

← Older revision		Revision as of 09:47, 11 December 2020
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	Given an [[~~EDO\|~~edo]] N and a positive rational number q, we may define the ''ambiguity'' ambig(N, q) of q in N edo by first computing u = N ~~log2~~(q), and from there v = abs(u - round(u)). Then ambig(N, q) = v/(1-v). Since v is a measure of the relative error of q in is best approximation in N edo, and 1-v of its second best approximation, ambig(N, q) is the ratio of the best approximation to the second best. If we used [[~~Relative_cent\|~~relative cent]]s instead to measure relative error, we would get the same result.		Given an [[edo]] ''N'' and a positive rational number ''q'', we may define the '''ambiguity''' ambig(''N'', ''q'') of ''q'' in ''N'' edo by first computing ''u'' = ''N'' log<sub>2</sub>(''q''), and from there ''v'' = abs(''u'' - round(''u'')). Then ambig(''N'', ''q'') = ''v''/(1 - ''v''). Since ''v'' is a measure of the relative error of ''q'' in is best approximation in ''N'' edo, and 1 - ''v'' of its second best approximation, ambig(''N'', ''q'') is the ratio of the best approximation to the second best. If we used [[relative cent]]s instead to measure relative error, we would get the same result.

	Given a finite set s of positive rational numbers, the maximum value of ambig(N, q) for all ~~q∈s~~ is the ''Pepper ambiguity'' of N with respect to s. If the set s is the L odd limit [[~~Tonality_diamond\|~~tonality diamond]], this is the L-limit Pepper ambiguity of N. Lists of N of decreasing Pepper ambiguity can be found on the On-Line Encyclopedia of Integer Sequences, [~~https~~:~~//oeis.org/A117554 https://oeis.org/~~A117554], [~~https~~:~~//oeis.org/A117555 https://oeis.org/~~A117555], [~~https~~:~~//oeis.org/A117556 https://oeis.org/~~A117556], [~~https://oeis.org/A117557 https~~:~~//oeis.org/~~A117557], [~~https~~:~~//oeis.org/A117558 https://oeis.org/~~A117558] ~~and~~ [~~https~~:~~//oeis.org/A117559 https://oeis.org/~~A117559]. We may also define the mean ambiguity for N with respect to s by taking the mean of ambig(N, q) for all members q of s.		Given a finite set s of positive rational numbers, the maximum value of ambig(''N'', ''q'') for all ''q'' ∈ ''s'' is the '''Pepper ambiguity''' of ''N'' with respect to ''s''. If the set ''s'' is the ''L'' odd limit [[tonality diamond]], this is the ''L''-limit Pepper ambiguity of ''N''. Lists of ''N'' of decreasing Pepper ambiguity can be found on the [[On-Line Encyclopedia of Integer Sequences]]:

			* [[OEIS: A117554]]
			* [[OEIS: A117555]]
			* [[OEIS: A117556]]
			* [[OEIS: A117557]]
			* [[OEIS: A117558]]
			* [[OEIS: A117559]]

			We may also define the mean ambiguity for ''N'' with respect to ''s'' by taking the mean of ambig(''N'', ''q'') for all members ''q'' of ''s''.

			[[Category:Theory]]
			[[Category:Measure]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

← Older revision		Revision as of 00:00, 17 July 2018
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	~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~		Given an [[EDO\|edo]] N and a positive rational number q, we may define the ''ambiguity'' ambig(N, q) of q in N edo by first computing u = N log2(q), and from there v = abs(u - round(u)). Then ambig(N, q) = v/(1-v). Since v is a measure of the relative error of q in is best approximation in N edo, and 1-v of its second best approximation, ambig(N, q) is the ratio of the best approximation to the second best. If we used [[Relative_cent\|relative cent]]s instead to measure relative error, we would get the same result.
	~~This is~~ an ~~imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~
	~~: This revision was by author~~ [[~~User:genewardsmith~~\|~~genewardsmith]] and made on <tt>2018-01-10 14:53:50 UTC</tt>.<br>~~
	~~: The original revision id was <tt>624694375</tt>.<br>~~
	~~: The revision comment was: <tt></tt><br>~~
	~~The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>~~
	~~<h4>Original Wikitext content:</h4>~~
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Given an [[edo]] N and a positive rational number q, we may define the //ambiguity// ambig(N, q) of q in N edo by first computing u = N log2(q), and from there v = abs(u - round(u)). Then ambig(N, q) = v/(1-v). Since v is a measure of the relative error of q in is best approximation in N edo, and 1-v of its second best approximation, ambig(N, q) is the ratio of the best approximation to the second best. If we used [[relative cent]]s instead to measure relative error, we would get the same result.

	Given a finite set s of positive rational numbers, the maximum value of ambig(N, q) for all q∈s is the //Pepper ambiguity// of N with respect to s. If the set s is the L odd limit [[tonality diamond]], this is the L-limit Pepper ambiguity of N. Lists of N of decreasing Pepper ambiguity can be found on the On-Line Encyclopedia of Integer Sequences, https://oeis.org/A117554, https://oeis.org/A117555, https://oeis.org/A117556, https://oeis.org/A117557, https://oeis.org/A117558 and https://oeis.org/A117559. We may also define the mean ambiguity for N with respect to s by taking the mean of ambig(N, q) for all members q of s.		Given a finite set s of positive rational numbers, the maximum value of ambig(N, q) for all q∈s is the ''Pepper ambiguity'' of N with respect to s. If the set s is the L odd limit [[Tonality_diamond\|tonality diamond]], this is the L-limit Pepper ambiguity of N. Lists of N of decreasing Pepper ambiguity can be found on the On-Line Encyclopedia of Integer Sequences, [https://oeis.org/A117554 https://oeis.org/A117554], [https://oeis.org/A117555 https://oeis.org/A117555], [https://oeis.org/A117556 https://oeis.org/A117556], [https://oeis.org/A117557 https://oeis.org/A117557], [https://oeis.org/A117558 https://oeis.org/A117558] and [https://oeis.org/A117559 https://oeis.org/A117559]. We may also define the mean ambiguity for N with respect to s by taking the mean of ambig(N, q) for all members q of s.
	~~</pre></div>~~
	~~<h4>Original HTML content:</h4>~~
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Pepper ambiguity</title></head><body>Given an <a class="wiki_link" href="/edo">edo</a> N and a positive rational number q, we may define the <em>ambiguity</em> ambig(N, q) of q in N edo by first computing u = N log2(q), and from there v = abs(u - round(u)). Then ambig(N, q) = v/(1-v). Since v is a measure of the relative error of q in is best approximation in N edo, and 1-v of its second best approximation, ambig(N, q) is the ratio of the best approximation to the second best. If we used <a class="wiki_link" href="/relative%20cent">relative cent</a>s instead to measure relative error, we would get the same result.<br />
	~~<br />~~
	Given a finite set s of positive rational numbers, the maximum value of ambig(N, q) for all q∈s is the <em>Pepper ambiguity</em> of N with respect to s. If the set s is the L odd limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a>, this is the L-limit Pepper ambiguity of N. Lists of N of decreasing Pepper ambiguity can be found on the On-Line Encyclopedia of Integer Sequences, <!-- ws:start:WikiTextUrlRule:5:https://oeis.org/A117554 --><a class="wiki_link_ext" href="https://oeis.org/A117554~~" rel="nofollow">https://oeis.org/A117554</a><!-- ws:end:WikiTextUrlRule:5 -->~~, ~~<!-- ws:start:WikiTextUrlRule:6:~~https://oeis.org/A117555 ~~--><a class="wiki_link_ext" href="~~https://oeis.org/A117555~~" rel="nofollow">https://oeis.org/A117555</a><!-- ws:end:WikiTextUrlRule:6 -->~~, ~~<!-- ws:start:WikiTextUrlRule:7:~~https://oeis.org/A117556 ~~--><a class="wiki_link_ext" href="~~https://oeis.org/A117556~~" rel="nofollow">https://oeis.org/A117556</a><!-- ws:end:WikiTextUrlRule:7 -->~~, ~~<!-- ws:start:WikiTextUrlRule:8:~~https://oeis.org/A117557 ~~--><a class="wiki_link_ext" href="~~https://oeis.org/A117557~~" rel="nofollow">https://oeis.org/A117557</a><!-- ws:end:WikiTextUrlRule:8 -->~~, ~~<!-- ws:start:WikiTextUrlRule:9:~~https://oeis.org/A117558 ~~--><a class="wiki_link_ext" href="~~https://oeis.org/A117558~~" rel="nofollow">https://oeis.org/A117558</a><!-- ws:end:WikiTextUrlRule:9 -->~~ and ~~<!-- ws:start:WikiTextUrlRule:10:~~https://oeis.org/A117559 ~~--><a class="wiki_link_ext" href="~~https://oeis.org/A117559~~" rel="nofollow">https://oeis.org/A117559</a><!-- ws:end:WikiTextUrlRule:10 -->~~. We may also define the mean ambiguity for N with respect to s by taking the mean of ambig(N, q) for all members q of s.~~</body></html></pre></div>~~

Wikispaces>genewardsmith: Imported revision 624694375 - Original comment:

2018-01-10T14:53:50Z

**Imported revision 624694375 - Original comment: **

New page

<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2018-01-10 14:53:50 UTC</tt>.<br>
: The original revision id was <tt>624694375</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Given an [[edo]] N and a positive rational number q, we may define the //ambiguity// ambig(N, q) of q in N edo by first computing u = N log2(q), and from there v = abs(u - round(u)). Then ambig(N, q) = v/(1-v). Since v is a measure of the relative error of q in is best approximation in N edo, and 1-v of its second best approximation, ambig(N, q) is the ratio of the best approximation to the second best. If we used [[relative cent]]s instead to measure relative error, we would get the same result.

Given a finite set s of positive rational numbers, the maximum value of ambig(N, q) for all q∈s is the //Pepper ambiguity// of N with respect to s. If the set s is the L odd limit [[tonality diamond]], this is the L-limit Pepper ambiguity of N. Lists of N of decreasing Pepper ambiguity can be found on the On-Line Encyclopedia of Integer Sequences, https://oeis.org/A117554, https://oeis.org/A117555, https://oeis.org/A117556, https://oeis.org/A117557, https://oeis.org/A117558 and https://oeis.org/A117559. We may also define the mean ambiguity for N with respect to s by taking the mean of ambig(N, q) for all members q of s.
</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Pepper ambiguity</title></head><body>Given an <a class="wiki_link" href="/edo">edo</a> N and a positive rational number q, we may define the <em>ambiguity</em> ambig(N, q) of q in N edo by first computing u = N log2(q), and from there v = abs(u - round(u)). Then ambig(N, q) = v/(1-v). Since v is a measure of the relative error of q in is best approximation in N edo, and 1-v of its second best approximation, ambig(N, q) is the ratio of the best approximation to the second best. If we used <a class="wiki_link" href="/relative%20cent">relative cent</a>s instead to measure relative error, we would get the same result.<br />
<br />
Given a finite set s of positive rational numbers, the maximum value of ambig(N, q) for all q∈s is the <em>Pepper ambiguity</em> of N with respect to s. If the set s is the L odd limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a>, this is the L-limit Pepper ambiguity of N. Lists of N of decreasing Pepper ambiguity can be found on the On-Line Encyclopedia of Integer Sequences, <a class="wiki_link_ext" href="https://oeis.org/A117554" rel="nofollow">https://oeis.org/A117554</a>, <a class="wiki_link_ext" href="https://oeis.org/A117555" rel="nofollow">https://oeis.org/A117555</a>, <a class="wiki_link_ext" href="https://oeis.org/A117556" rel="nofollow">https://oeis.org/A117556</a>, <a class="wiki_link_ext" href="https://oeis.org/A117557" rel="nofollow">https://oeis.org/A117557</a>, <a class="wiki_link_ext" href="https://oeis.org/A117558" rel="nofollow">https://oeis.org/A117558</a> and <a class="wiki_link_ext" href="https://oeis.org/A117559" rel="nofollow">https://oeis.org/A117559</a>. We may also define the mean ambiguity for N with respect to s by taking the mean of ambig(N, q) for all members q of s.</body></html></pre></div>

← Older revision		Revision as of 09:48, 11 December 2020
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	We may also define the mean ambiguity for ''N'' with respect to ''s'' by taking the mean of ambig(''N'', ''q'') for all members ''q'' of ''s''.		We may also define the mean ambiguity for ''N'' with respect to ''s'' by taking the mean of ambig(''N'', ''q'') for all members ''q'' of ''s''.

	[[Category:~~Theory~~]]		[[Category:EDO theory pages]]
			[[Category:Terms]]
	[[Category:Measure]]		[[Category:Measure]]